A333947 a(n) is the smallest k > 0 such that sigma(n+k) = sigma(n); if such k > 0 does not exist, then a(n) = 0.

0, 0, 0, 0, 0, 5, 0, 0, 0, 7, 0, 0, 0, 1, 8, 9, 0, 0, 0, 6, 10, 0, 0, 14, 0, 15, 0, 11, 0, 16, 0, 0, 2, 19, 12, 0, 0, 21, 0, 18, 0, 20, 0, 21, 0, 5, 0, 27, 0, 0, 4, 45, 0, 2, 16, 31, 22, 31, 0, 18, 0, 7, 40, 0, 18, 4, 0, 14, 8, 24, 0, 0, 0, 39, 0, 63, 0, 14, 0

Offset: 1

Author

Bernard Schott, Apr 11 2020

Keywords

• nonn

Comments

This sequence is inspired by A007365 where a(n) is the smallest k such that sigma(n+k) = sigma(k); indeed, n and k are switched between these two sequences.

There are three distinct cases for which a(n) = 0:

If n is prime then a(n) = 0,

If n is in A211658 then a(n) = 0,

If n is the largest number q_r of a sequence q_1 < q_2 < ... < q_r with q_r composite and sigma(q_1) = sigma(q_2) = ... = sigma(q_r) then a(n) = 0. The first two such examples are a(25) = 0 and a(39) = 0 with sigma(16) = sigma(25) = 31, and sigma(28) = sigma(39) = 56.

Examples

			sigma(9) = 13 and there is no k>0 such that sigma(9+k) = 13, then a(9) = 0.
sigma(14) = sigma(15) = sigma(23) = 24, so a(14) = 1 and a(15) = 8, and as 23 is prime, a(23) = 0.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems

Crossrefs

Cf. A000203, A007365, A054973.

Cf. A002961 (a(n)=1).

Programs

Maple

f:= proc(n) local s,k;
  s:= numtheory:-sigma(n);
for k from n+1 to s-1 do
  if numtheory:-sigma(k)=s then return k-n fi
od;
0
end proc:
map(f, [$1..100]); # Robert Israel, Apr 17 2020

Mathematica

a[n_] := Module[{k = n+1, s = DivisorSigma[1, n]}, While[k < s && DivisorSigma[1, k] != s, k++];If[k >= s, 0, k-n]]; Array[a, 70] (* Amiram Eldar, Apr 12 2020 *)

PARI

a(n) = {my(s=sigma(n)); for (k= n+1, s-1, if (sigma(k) == s, return (k-n));); return(0);} \\ Michel Marcus, Apr 11 2020